• Blended Learning & Flipped Classroom

The goal of the course is to provide  students with the theoretical knowledge of the basic numerical linear algebra techniques and to let them develop the ability to apply such knowledge to relevant model problems. The course covers numerical methods for determined and overdetermined linear systems, their extension to nonlinear systems and their application to least square problems. All topics are presented at a theoretical level and the corresponding methods are applied to the solution of numerical problems in the computer laboratory sessions based on the use of MATLAB scientific software.

Knowledge and understanding (DD1)

The students will acquire detailed knowledge and understanding of the basic principles of numerical computation of the solution of linear and nonlinear system.

Apply knowledge and understanding (DD2)

The students will be able to apply this knowledge to the solution of relevant numerical probems in linear algebra and least square problems, making use of the resources of the MATLAB scientific software.

Making judgements (DD3)

The students will be able to make judgements on the most appropriate numerical method to be applied for the solution of different classes of problems in terms of accuracy and efficiency.

Communications skills (DD4)

The students will be able to present the results of the numerical analysis of a linear algebra problem with appropriate technical language and to understand mathematical proofs presented in applied mathematics textbooks.

1) Review of linear algebra concepts. Introduction to the scientific software MATLAB.
Introduction to the floating point representation of real numbers.

2) Numerical solution of determined and linear systems by direct methods: Gaussian elimination, LU factorization, Cholesky factorization. Stability analysis for the numerical solution of linear systems. Condition number of a matrix.

3) Basic concepts on the Newton method for nonlinear systems.

4) Basic iterative methods and sparse matrices.

5) Numerical methods for overdetermined linear systems: QR factorization, SVD decomposition.

6) Linear and nonlinear least square problems, regularization of ill posed problems.

Students must have good theoretical knowledge of all the results on real and complex number theory, mathematical analysis of real and vector valued functions, analytic geometry and basic linear algebra theory that are usually included in Engineering Bachelor courses at Politecnico di Milano.

The assessment will consist in an individual written assessment, based on the solution of a set of numerical linear algebra problems with the help of the MATLAB scientific software. In this written test the students will have to demonstrate

a) the acquired ability to apply numerical methods for the solution of relevant linear algebra and least square problems

b) the ability to make judgements on the most appropriate numerical method to be applied for the solution of different classes of problems in terms of accuracy and efficiency

c) the ability to present the results of the numerical analysis of a linear algebra problem with appropriate technical language

d) the ability to present the basic theoretical concepts introduced in the course with appropriate technical language

e) the acquired knowledge and understanding of the basic principles of numerical computation of the solution of linear and nonlinear system.

The written test can be complemented by an optional individual oral interview, in which students will have the possibility to further demonstrate

a) the ability to present the basic theoretical concepts introduced in the course with appropriate technical language

b) the acquired knowledge of the basic principles of numerical computation of the solution of linear and nonlinear system,

in case they believe they had not succeeded  in doing so properly in the written examination only.